Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<iostream>
#include<stdio.h>
using namespace std;
double DP[3010][3010];
double X[3010];
int main()
{
  int N;
  cin>>N;
  for(int i=0;i<N;i++)
    cin>>X[i];
  DP[0][0]=1;
  for(int i=0;i<N;i++)
    for(int j=0;j<=i;j++)
      DP[i+1][j+1]+=DP[i][j]*X[i],
      DP[i+1][j]+=DP[i][j]*(1-X[i]);
  double ans=0;
  for(int i=(N+1)/2;i<=N;i++)
    ans+=DP[N][i];
  printf("%.9lf\n",ans);
  return 0;
}